The exchange lemma and Gaussian elimination

Thanks to this comment, I have finally decided to try to understand in what sense Gaussian elimination and the Steinitz exchange lemma are “basically the same thing”. It’s not at all hard to spot similarities, but it seems to be a little trickier to come up with a purely mechanical process for translating proofs in one language into proofs in the other.

It might be of some interest to know how I approached this post. Rather than working everything out in advance, I started with an incomplete understanding of the connection. I had thought about it enough to convince myself that I could get to the end, but found that as I proceeded there were a few surprises, and the eventual connection was not quite as close as I was expecting. (Actually, this paragraph is slightly misleading. I am writing it while in the middle of writing the rest of the post. I’ve had a few surprises, and though I am fairly sure I’ll get to the end I am not quite sure what the end will look like. [Note added after I'd got to the end: it was nothing like what I expected.])

One way of looking at Cauchy’s theorem

Cauchy’s theorem is the assertion that the path integral of a complex-differentiable function around a closed curve is zero (as long as there aren’t any holes inside the curve where the function has singularities or isn’t defined). This theorem, which is fundamental to complex analysis, can be vastly generalized and seen from many different points of view. This post is about a little idea that occurred to me once when I was teaching complex analysis. I meant to include it as a Mathematical Discussion on my web page but didn’t get round to it. Now, while the novelty of having a blog still hasn’t worn off, I find I have the energy to put it here.

Let’s start with a simpler fact: that if is a function from to and the derivative of is everywhere zero, then is constant. What is the natural generalization of this fact to functions defined on the plane?

Discovering a formula for the cubic

In this post I want to revisit a topic that I first discussed on my web page here. My aim was to present a way in which one might discover a solution to the cubic without just being told it. However, the solution that arose was not very nice, and at the end I made the comment that I did not know a way of removing the rabbit-out-of-a-hat feeling that the usual much neater formula for the cubic (together with its derivation) left me with.

A couple of years ago, I put that situation right by stumbling on a very simple idea about quadratic equations that generalizes easily to cubics. More to the point, the stumble wasn’t completely random, so the entire approach can be justified as the result of standard and easy research strategies. I am no historian, but I would imagine that this idea is pretty similar to the idea (in some equivalent form) that first led to this solution.